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Math Help - BMO 2006/7 circles question

  1. #1
    Senior Member abhishekkgp's Avatar
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    BMO 2006/7 circles question

    Two touching circles S and T share a common tangent which meets
    S at A and T at B. Let AP be a diameter of S and let the tangent
    from P to T touch it at Q. Show that AP = PQ.

    If you have solved it then please give me a hint(don't post the full solution).
    Thanks in advance.
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  2. #2
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    earboth's Avatar
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    Re: BMO 2006/7 circles question

    Quote Originally Posted by abhishekkgp View Post
    Two touching circles S and T share a common tangent which meets
    S at A and T at B. Let AP be a diameter of S and let the tangent
    from P to T touch it at Q. Show that AP = PQ.

    If you have solved it then please give me a hint(don't post the full solution).
    Thanks in advance.
    1. Draw a sketch!

    2. Prove(!) that the described situation is only possible if the 2 circles are congruent.

    3. Prove that the 2 diameters and the tangent segments consequently form a square.
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  3. #3
    Senior Member abhishekkgp's Avatar
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    Re: BMO 2006/7 circles question

    Quote Originally Posted by earboth View Post
    1. Draw a sketch!

    2. Prove(!) that the described situation is only possible if the 2 circles are congruent.

    3. Prove that the 2 diameters and the tangent segments consequently form a square.
    I have attached a sketch. The two circle S and T drawn are not congruent but i still get AP=PQ.
    Attached Thumbnails Attached Thumbnails BMO 2006/7 circles question-img_0040.jpg  
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